The discovery of the \(X(3872)\)  [also known as \(\chi _{c1}(3872)\) ] resonance by the Belle Collaboration in 2003 [1]} and confirmed later by other experiments [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, ushered in a new era in hadron physics [11]}, [12]}, [13]}, which cannot be easily explained as a conventional charmonium state in quark models [14]}, [15]}. The isospin of \(X(3872)\)  is zero, and its quantum numbers \(J^{PC} = 1^{++}\)  were extracted by the LHCb Collaboration [9]}, [10]}. Particularly, the mass of \(X(3872)\) , \(M_{X(3872)} = 3871.65 \pm 0.06\)  MeV, is very close to the threshold of \(D^0\bar{D}^{\ast 0}\) , \(M_{D^0\bar{D}^{\ast 0}} = 3871.69 \pm 0.11\)  MeV, and its width is \(\Gamma _{X(3872)} = 1.19 \pm 0.21\)  MeV [18]}, which is rather narrow. A precise determination of the mass of the \(X(3872)\)  resonance is crucial to understanding its nature [19]}. Furthermore, the branching fractions of the \(X(3872)\)  resonance decaying into \(J/\psi \rho ^0\)  and \(J/\psi \omega \)  are similar [20]}. It is found that the contributions of the \(\omega \)  meson to the \(X(3872) \rightarrow J/\psi \pi ^+ \pi ^-\)  decays are sizeable [20]}, [22]}, which indicates that the \(X(3872)\)  resonance has sizeable coupling to the \(J/\psi \omega \)  channel. Recently, the \(e^+ e^- \rightarrow \omega X(3872)\)  reaction was observed by the BESIII Collaboration [23]}.
